Answer :
To determine which of the given expressions are like radicals, we need to identify which radicals have the same expression under the radical sign.
### Given Radicals:
1. [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
2. [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
3. [tex]\( 4 \sqrt[3]{6 a b^2} \)[/tex] (a cube root expression, not a square root)
4. [tex]\( 2 \sqrt{6 b^2 a} \)[/tex] (which is the same as [tex]\( 2 \sqrt{6 a b^2} \)[/tex] due to commutative property of multiplication)
### Step-by-Step Process:
- Step 1: Identify the type of root
- The first expression has a square root: [tex]\( \sqrt{6 a b^2} \)[/tex]
- The second expression has a square root: [tex]\( \sqrt{6 a b^2} \)[/tex]
- The third expression has a cube root: [tex]\( \sqrt[3]{6 a b^2} \)[/tex]
- The fourth expression has a square root: [tex]\( \sqrt{6 b^2 a} \)[/tex]
- Step 2: Simplify the order of the terms under the radical (if necessary) to make comparisons easier
- The fourth expression, [tex]\( 2 \sqrt{6 b^2 a} \)[/tex], can be rewritten as [tex]\( 2 \sqrt{6 a b^2} \)[/tex] because the order of multiplication does not matter.
- Step 3: Compare the radicals (ignoring the coefficients outside the square root)
- [tex]\( \sqrt{6 a b^2} \)[/tex] is the radical part common in the first expression.
- [tex]\( \sqrt{6 a b^2} \)[/tex] is the radical part common in the second expression.
- [tex]\( \sqrt{6 a b^2} \)[/tex] (reordered from [tex]\( \sqrt{6 b^2 a} \)[/tex]) is the radical part common in the fourth expression.
- [tex]\( \sqrt[3]{6 a b^2} \)[/tex] differs because it is a cube root, not a square root.
### Conclusion:
The radicals that are like radicals (having the same expression inside the square root, irrespective of coefficients) are:
- [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- [tex]\( 2 \sqrt{6 a b^2} \)[/tex] (formerly written as [tex]\( 2 \sqrt{6 b^2 a} \)[/tex])
Therefore, the like radicals are:
- [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- [tex]\( 2 \sqrt{6 a b^2} \)[/tex]
### Given Radicals:
1. [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
2. [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
3. [tex]\( 4 \sqrt[3]{6 a b^2} \)[/tex] (a cube root expression, not a square root)
4. [tex]\( 2 \sqrt{6 b^2 a} \)[/tex] (which is the same as [tex]\( 2 \sqrt{6 a b^2} \)[/tex] due to commutative property of multiplication)
### Step-by-Step Process:
- Step 1: Identify the type of root
- The first expression has a square root: [tex]\( \sqrt{6 a b^2} \)[/tex]
- The second expression has a square root: [tex]\( \sqrt{6 a b^2} \)[/tex]
- The third expression has a cube root: [tex]\( \sqrt[3]{6 a b^2} \)[/tex]
- The fourth expression has a square root: [tex]\( \sqrt{6 b^2 a} \)[/tex]
- Step 2: Simplify the order of the terms under the radical (if necessary) to make comparisons easier
- The fourth expression, [tex]\( 2 \sqrt{6 b^2 a} \)[/tex], can be rewritten as [tex]\( 2 \sqrt{6 a b^2} \)[/tex] because the order of multiplication does not matter.
- Step 3: Compare the radicals (ignoring the coefficients outside the square root)
- [tex]\( \sqrt{6 a b^2} \)[/tex] is the radical part common in the first expression.
- [tex]\( \sqrt{6 a b^2} \)[/tex] is the radical part common in the second expression.
- [tex]\( \sqrt{6 a b^2} \)[/tex] (reordered from [tex]\( \sqrt{6 b^2 a} \)[/tex]) is the radical part common in the fourth expression.
- [tex]\( \sqrt[3]{6 a b^2} \)[/tex] differs because it is a cube root, not a square root.
### Conclusion:
The radicals that are like radicals (having the same expression inside the square root, irrespective of coefficients) are:
- [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- [tex]\( 2 \sqrt{6 a b^2} \)[/tex] (formerly written as [tex]\( 2 \sqrt{6 b^2 a} \)[/tex])
Therefore, the like radicals are:
- [tex]\( 9 \sqrt{6 a b^2} \)[/tex]
- [tex]\( -5 \sqrt{6 a b^2} \)[/tex]
- [tex]\( 2 \sqrt{6 a b^2} \)[/tex]
